The classification of topological states of matter is an important hot topic in mathematical physics. In this talk I will describe a new approach to the classification of topological quantum systems in class AIII which is based on the study of a new category of vector bundles. The objects of this category, the chiral vector bundles, are pairs constituted by a complex vector bundle along with one of its automorphisms. We provide a classification for the homotopy equivalence classes of these objects which is based on the construction of a suitable classifying space. The computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: Some of those just reproduce the ordinary Chern classes but there are also new odd-dimensional classes which take care of the extra topological information introduced by the chiral structure. Chiral vector bundles provide the proper geometric model for topological quantum systems in class AIII, namely for systems endowed with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles over sphere and tori (explicitly computable up to dimension 4) recover the commonly accepted classification for topological insulators of class AIII which is usually based on the K-group K1. However, this new classification turns out to be even richer since it takes care also for the possibility of non trivial Chern classes.